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Math Help - Vector field

  1. #1
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    Vector field

    F(x,y) = - \frac{y}{x^2+y^2}i + \frac{x}{x^2+y^2}j

    1) What domain of the field?

    2) Prove that:
    \oint_{\alpha} - \frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy = 2 \pi

    \alpha is a circumference of center in origin and raio r

    3) Check the green's theorem that:
    \oint_{\alpha} - \frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy = 0

    4) Explain the contradiction


    My solution:

    1) IR- \{0,0\}

    2) \int_0^{2 \pi} (sen^2t + cos^2t)dt = 2 \pi

    3) \frac{ \partial (\frac{x}{x^2+y^2})}{ \partial x} - \frac{ \partial (\frac{y}{x^2+y^2})}{ \partial y} = 0

    \int \int_S 0 dxdy = 0

    4) As the domain is IR-\{0,0\} not is possible use the theorem of green

    All this correct ?
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  2. #2
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    Please it is correct ?
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  3. #3
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    Quote Originally Posted by Apprentice123 View Post
    F(x,y) = - \frac{y}{x^2+y^2}i + \frac{x}{x^2+y^2}j

    1) What domain of the field?

    2) Prove that:
    \oint_{\alpha} - \frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy = 2 \pi

    \alpha is a circumference of center in origin and raio r

    3) Check the green's theorem that:
    \oint_{\alpha} - \frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy = 0

    4) Explain the contradiction


    My solution:

    1) IR- \{0,0\}

    2) \int_0^{2 \pi} (sen^2t + cos^2t)dt = 2 \pi

    3) \frac{ \partial (\frac{x}{x^2+y^2})}{ \partial x} - \frac{ \partial (\frac{y}{x^2+y^2})}{ \partial y} = 0

    \int \int_S 0 dxdy = 0

    4) As the domain is IR-\{0,0\} not is possible use the theorem of green

    All this correct ?
    Very good! If \int_c P\,dx + Q\, dy = \iint_R Qx-Py \,dA then to use Green's theorem requires that P and Q are continuous and have continuous partial derivatives in R which as you stated is not true. As classic problem is to show that for your vector field that

     <br />
\int_c P\,dx + Q\, dy = 2 \pi<br />

    for any simple closed curve C enclosing the origin.
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  4. #4
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    thank you
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